Abstract. We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups GL(n) and SL(n) we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur's trace formula obtained in [FLM11,FL11], which allows us to show that for GL(n) and SL(n) the contribution of the continuous spectrum is negligible in the limit.

We derive a refinement of the spectral expansion of Arthur's trace formula. The expression is absolutely convergent with respect to the trace norm.

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H 1 (Γ, En), where Γ is a lattice in SL(2, C) and En = Sym n ⊗Sym n , n ∈ N ∪ {0}, is one of the standard self-dual modules. In the case Γ = SL(2, O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2, O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n 2 / log n) for any fixed lattice Γ in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.

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The motivating question of this paper is roughly the following: given a flat group scheme G over Z p , p prime, with semisimple generic fiber G Qp , how far are open subgroups of G(Z p ) from subgroups of the form X(Z p )K p (p n ), where X is a subgroup scheme of G and K p (p n ) is the principal congruence subgroup Ker(G(Z p ) → G(Z/p n Z))? More precisely, we will show that for G Qp simply connected there exist constants J ≥ 1 and ε > 0, depending only on G, such that any open subgroup of G(Z p ) of level p n admits an open subgroup of index ≤ J which is contained in X(Z p )K p (p ⌈εn⌉ ) for some proper, connected algebraic subgroup X of G defined over Q p . Moreover, if G is defined over Z, then ε and J can be taken independently of p.We also give a correspondence between natural classes of Z p -Lie subalgebras of g Zp and of closed subgroups of G(Z p ) that can be regarded as a variant over Z p of Nori's results on the structure of finite subgroups of GL(N 0 , F p ) for large p [Nor87].As an application we give a bound for the volume of the intersection of a conjugacy class in the group G(Ẑ) = p G(Z p ), for G defined over Z, with an arbitrary open subgroup. In a companion paper, we apply this result to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice G(Z).